On the calculation of magnetic fields generated by coils
E. Kasper
Aug 1, 1995
Citations
3
Citations
Journal
Journal of Microscopy
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Key Takeaway Key Takeaway: The BEM method can effectively calculate magnetic fields produced by coils, but it is necessary to apply a special theory to overcome the inefficiency of Gauss quadratures.
Abstract
The design of unconventional magnetic lenses, which was performed and presented by Tom Mulvey and his coworkers (Mulvey & Nasr. 1981: Mulvey, 1982, 1984; Tahir & Mulvey. 1990). brought considerable improvements to scanning electron microscopes. Moreover, but not so obviously, these designs were a permanent challenge to develop new techniques for numerical field calculation. For example, the open structures of snorkel lenses require considerable memory, if treated with the conventional finite-element method (FEM), since the field-confining boundaries must be located quite far from the yoke. This difficulty was investigated and described in detail by Mulvey (1982). Various attempts have been made to overcome these difficulties. One of them is presented in Fig 1; this shows a triangulation of a cross-section through a magnetic lens, as was designed by Mulvey. The idea was to apply the FEM inside the yoke and the boundary-element method (BEM) outside: the fields on both sides of the yoke surface must then be coupled by the well-known continuity laws. In principle this method should work, but the investigations remained incomplete. Stroer (1987, 1990) and Kasper & Stroer (1990) found a very efficient. version of the BEM and were able to overcome all these difficulties. The BEM was applied to various types of I.;nses, and Mulvey's designs gave excellent examples to test this method. Stroer (1987, 1990) could calculate the magnetic field throughout the space with high accuracy; however, the yoke must be assumed to be unsaturated. This method, which has been described in detail elsewhere, will not be discussed here. One particular difficulty in the BEM, which was noted by Stroer (1987, 1990) and overcome by a special theory, is, surprisingly, the calculation of the magnetic fields produced by coils, even in the absence of any ferromagnetic material. In principle this can be carried out entirely by numerical integration using the familiar Biot-Savart law. But it soon small, so that Gauss quadratures become inefficient. It is thus obvious that a considerable gain in computation speed would be achieved, if it were possible to find surface sources that produce the same field outside the coil as does the spatial current distribution. One method for this task was presented by Stroer (1987). but this requires the solution of an additional Dirichlet problem. Another way. which leads directly to simple explicit formulae for such substitution sources, is the subject of this paper.